Computing Sensitivities for Distortion Risk Measures

Distortion risk measure, defined by an integral of a distorted tail probability, has been widely used in behavioral economics and risk management as an alternative to expected utility. The sensitivity of the distortion risk measure is a functional of certain distribution sensitivities. We propose a new sensitivity estimator for the distortion risk measure that uses generalized likelihood ratio estimators for distribution sensitivities as input and establish a central limit theorem for the new estimator. The proposed estimator can handle discontinuous sample paths and distortion functions.

OPTIMAL INSURANCE CONTRACTS UNDER DISTORTION RISK MEASURES WITH AMBIGUITY AVERSION

This paper presents analytical representations for an optimal insurance contract under distortion risk measure and in the presence of model uncertainty. We incorporate ambiguity aversion and distortion risk measure through the model of Robert and Therond [(2014) ASTIN Bulletin: The Journal of the IAA, 44(2), 277–302.] as per the framework of Klibanoff et al. [(2005) A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849–1892.]. Explicit optimal insurance indemnity functions are derived when the decision maker (DM) applies Value-at-Risk as risk measure and is ambiguous about the loss distribution. Our results show that: (1) under model uncertainty, ambiguity aversion results in a distorted probability distribution over the set of possible models with a bias in favor of the model which yields a larger risk; (2) a more ambiguity-averse DM would demand more insurance coverage; (3) for a given budget, uncertainties about the loss distribution result in higher risk level for the DM.

RISK REDISTRIBUTION GAMES WITH DUAL UTILITIES

This paper studies optimal risk redistribution between firms, such as institutional investors, banks or insurance companies. We consider the case where every firm uses dual utility (also called a distortion risk measure) to evaluate risk. We characterize optimal risk redistributions via four properties that need to be satisfied jointly. The characterized risk redistribution is unique under three conditions. Whereas we characterize risk redistributions by means of properties, we can also use some results to study competitive equilibria. We characterize uniqueness of the competitive equilibrium in markets with dual utilities. Finally, we identify two conditions that are jointly necessary and sufficient for the case that there exists a trade that is welfare-improving for all firms.